# 세미나 및 콜로퀴엄

구글 Calendar나 iPhone 등에서 구독하면 세미나 시작 전에 알림을 받을 수 있습니다.

Millions of Korean workers follow night or nonstandard shifts. A similar number of Koreans travel overseas each year. This leads to misalignment of the daily (circadian) clock in individuals: a clock that controls sleep, performance, and nearly every physiological process in our body. A mathematical model of this clock can be used to optimize schedules to maximize productivity and minimize jetlag. We simulate this model in a smartphone app, ENTRAIN (www.entrain.org), which has been installed in phones over 200,000 times in over 100 countries. I will discuss techniques we have been developing and clinically testing to determine sleep stage and circadian time from wearable data, for example, as collected by our app or used in many commercially available sleep trackers. This project has led us to develop new techniques to: 1) estimate phase from noisy data with gaps, 2) rapidly simulate of population densities from high dimensional models and 3) determine how mathematical models of sleep and circadian physiology can be used with machine learning techniques to improve predictions.

(This is a reading seminar for graduate students.)

Grothendieck's $K_0$-group has an exact sequence $K_0(Z)\to K_0(X)\to K_0(X-Z)\to 0$ for a closed immersion $Z\to X$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $K$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $Q$-construction of an exact category, higher $K$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $K_0$ and relates higher K-theory with Chow groups. This is the first half of Quillen's algebraic $K$-theory.

Finite element discretization of solutions with respect to simplicial/cubical meshes has been studied for decades, resulting in a clear understanding of both the relevant mathematics and computational engineering challenges. Recently, there has been both a desire and need for an equivalent body of research regarding discretization with respect to generic polygonal/polytopal meshes. General meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and non-matching interfaces. Such a flexibility represents a powerful tool towards the efficient solution of problems with complex inclusions as in geophysical applications or posed on very complicated or possibly deformable geometries as encountered in basin and reservoir simulations, in fluid-structure interaction, crack propagation or contact problems.

In this talk, a new computational paradigm for discretizing PDEs is presented via staggered Galerkin approach on general meshes. First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal meshes for elliptic problems are proposed. The method can be flexibly applied to rough grids such as highly distorted meshes. Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging nodes.

Let $X$ be an abelian variety of dimension $g$ over a field $k$. In general, the group $textrm{Aut}_k(X)$ of automorphisms of $X$ over $k$ is not finite. But if we fix a polarization $mathcal{L}$ on $X$, then the group $textrm{Aut}_k(X,mathcal{L})$ of automorphisms of the polarized abelian variety $(X,mathcal{L})$ over $k$ is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over $k.$

In this talk, we give a classification of such finite groups for the case when $k$ is a finite field and $g$ is a prime number. If $g=2,$ then we need a notion of maximality in a certain sense, and for $g geq 3,$ we achieve a rather complete list without conveying maximality.

(This is a reading seminar for graduate students.)

Grothendieck's $K_0$-group has an exact sequence $K_0(Z)\to K_0(X)\to K_0(X-Z)\to 0$ for a closed immersion $Z\to X$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $K$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $Q$-construction of an exact category, higher $K$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $K_0$ and relates higher K-theory with Chow groups. This is the second half of Quillen's algebraic $K$-theory.

Recently, the classification of isoparametric hypersurfaces in spheres has been completed. Therefrom, various new research projects in geometry have been initiated. The study of minimal lagrangian submanifolds via isoparametric hypersurfaces is one of the most active projects a la mode. In this talk, we have an introduction to the study of Isoparametric hypersurfaces and minimal Lagrangian submanfolds, and discuss the relationship between them.

(This is a reading seminar for graduate students.)

Algebraic K-theory originated with the Grothendieck-Riemann-Roch theorem, a generalization of Riemann-Roch theorem to higher dimensional varieties. For this, we shall discuss the definitions of $K_0$-theory of a variety, its connection with intersection theory, $\lambda$-operation, $\gamma$-filtration, Chern classes and Adams operations.

I report on work with M. Gubinelli and T. Oh on the

renormalized nonlinear wave equation

in 2d with monomial nonlinearity and in 3d with quadratic nonlinearity.

Martin Hairer has developed an efficient machinery to handle elliptic

and parabolic problems with additive white noise, and many local

existence questions are by now well understood. In contrast not much is

known for hyperbolic equations. We study the simplest nontrivial

examples and prove local existence and weak universality, i.e. the

nonlinear wave equations with additive white noise occur as scaling

limits of wave equations with more regular noise.

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IBS, Room B232 (DIMAG)
Discrete Math
Andreas Holmsen (KAIST)
Large cliques in hypergraphs with forbidden substructures

A result due to Gyárfás, Hubenko, and Solymosi, answering a question of Erdős, asserts that if a graph G does not contain K2,2 as an induced subgraph yet has at least c(n2) edges, then G has a complete subgraph on at least c210n vertices. In this paper we suggest a “higher-dimensional” analogue of the notion of an induced K2,2, which allows us to extend their result to k-uniform hypergraphs. Our result also has interesting consequences in topological combinatorics and abstract convexity, where it can be used to answer questions by Bukh, Kalai, and several others.